While writing a textbook for an upper-level math class, a Franklin & Marshall College professor discovered that he has learned as much from his student assistants as they have from him.
"Their insightful questions and ideas have driven me to write the best book I was capable of writing," said Professor of Mathematics Robert Gethner, whose textbook is designed for the College's "Introduction to Real Analysis" course.
The two students helping Gethner on the book, tentatively titled "Mathematical Analysis and the Real Numbers: A Beginner's Guide to their Wonders," are seniors Andrew Miller and Donghui Li. Miller is using mathematical software to create diagrams, and Li is proofreading and suggesting changes and alternative approaches.
Gethner said his vision for the textbook evolved as he taught the course over the last eight years. He observed topics that excited students and sought ways to make mathematical analysis's difficult concepts clear to his students.
"I've learned a lot about the foundations of my specialty in the process of working on the book,” he said.
Gethner sent a draft of the book to a publisher, the Mathematical Association of America, last December. An editor there was enthusiastic about the project and encouraged him to revise and resubmit. Gethner said he intends to send a finished manuscript this spring.
"Mathematical Analysis" centers on the deep logical analysis of the mechanics of calculus, Gethner said. His work explains abstract ideas to students in a conversational writing style, untangling intricate mathematical constructions in more detail than most other books on the subject, he said.
"As a writer, I'm trying to create the illusion that I'm in the room with my reader, chatting with them, telling them an engaging story and answering their questions, while at the same time being mathematically precise," he said. "My goal is to invite students inside the subject."
Miller and Li have been instrumental in this effort, said Gethner, who is chair of the Mathematics Department. Miller started working with the professor last spring. Li joined in the fall. The students work independently and meet weekly with Gethner.
Miller, a philosophy minor, said math became exciting for him when he took the department's "Introduction to Higher Mathematics."
"It was the first time that I got to see that mathematical thinking is closely related to the way we think in philosophy," he said. Miller produces computer-generated diagrams to illustrate the abstract theorems Gethner presents in the textbook.
“He is great at finding clever ways to make the software produce what we need,” Gethner said.
As proofreader, Li "notices all kinds of valuable things," Gethner said. He took note of Li's editorial ability during a course he was teaching that used an earlier version of his book.
"She would come into my office to ask questions about the assigned reading," he recalled. "It was clear she had read very thoroughly. I would look at what I’d written and find myself thinking, 'What was this guy trying to say?'"
Li said she likes that Gethner uses metaphors to explain abstract ideas.
For example, to clarify the idea of a "dense set," Gethner used the analogy of too many students attending a party in a crowded house, forcing some to move outside to alleviate the pressure inside.
"It actually helped me to understand the abstract idea better," Li said.
Li and Miller said the key benefits of working on the textbook have been a greater understanding of the subject and the opportunity to work with a professor on his scholarship.
"The beauty of Li and Miller's work is that not only are they a huge help to me, but they're also learning from the experience," Gethner said.